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Amoebas, Monge-Amp`Ere Measures, and Triangulations of the Newton

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Amoebas, Monge-Amp`Ere Measures, and Triangulations of the Newton AMOEBAS, MONGE-AMPERE` MEASURES, AND TRIANGULATIONS OF THE NEWTON POLYTOPE MIKAEL PASSARE and HANS RULLGARD˚ Abstract The amoeba of a holomorphic function f is, by definition, the image in Rn of the zero locus of f under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function N f with the property that its Monge-Ampere` mass is concentrated to the amoeba of f . We obtain results of two kinds; by approximating N f with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of f ; by computing the Monge-Ampere` measure, we find sharp bounds for the area of amoebas 2 in R . We also consider systems of functions f1,..., fn and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations. 1. Introduction The classical Jensen formula can be regarded as a relation between the zeros of a holomorphic function f and the properties of a certain convex function associated to f . In this paper we consider an analogue of this convex function, which we denote N f , in the case where f is a function of several variables, and we search for relations with the zero locus of f . The results we find are of two kinds. In Section 3, we show that approximation of N f by a piecewise linear function leads to an approximation of the so-called amoeba by a polyhedral complex (Th. 1). When f is a polynomial, this polyhedral complex is dual (in a precise sense) to a certain subdivision of the Newton polytope of f . An explicit computation of the polyhedral complex amounts to evaluating certain param- eters depending on the coefficients of f . It turns out that these satisfy a GKZ-type system of differential equations and can be expanded in a hypergeometric power se- ries involving the coefficients (Th. 3). DUKE MATHEMATICAL JOURNAL Vol. 121, No. 3, c 2004 Received 18 September 2002. Revision received 3 April 2003. 2000 Mathematics Subject Classification. Primary 32A60; Secondary 52A41, 52B20. 481 482 PASSARE and RULLGARD˚ In Section 5 we investigate the measures arising when Monge-Ampere` or Laplace operators act on N f . We find several relations between such measures and the hyper- surface defined by f (Ths. 5 and 6). The former can be regarded as a local variant of the Bernstein formula for the number of solutions to a system of polynomial equa- tions. We also obtain an estimate on the Monge-Ampere` measure, which gives an upper bound on the area of the amoeba of a polynomial in two variables (Th. 7 and Cor. 1). This estimate was used in [8] to give a new characterization of so-called Har- nack curves, which are fundamental for Hilbert’s sixteenth problem in real algebraic geometry. 2. Background Suppose that f is an entire function in the complex plane with zeros a1, a2, a3,... ordered so that |a1| ≤ |a2| ≤ · · · . Assume for simplicity that f (0) 6= 0. The Jensen formula states that m 1 Z 2π X r log | f (reit )| dt = log | f (0)| + log , 2π |ak| 0 k=1 where m is the largest index such that |am| < r. If this expression is considered as a function N f of log r, there is a strong connection between this function and the zeros of f . Thus it follows immediately that N f is a piecewise linear convex function whose gradient is equal to the number of zeros of f inside the disc {|z| < r}. The second derivative of N f , in the sense of distributions, is a sum of point masses at log |ak|, k = 1, 2, 3,.... In this paper we consider a certain generalization of the function occurring in the Jensen formula to holomorphic functions of several variables. Let be a convex open set in Rn, and let f be a holomorphic function de- −1 n n fined in Log (), where Log : (C \{0}) −→ R is the mapping (z1,..., zn) 7→ (log |z1|,..., log |zn|). In [12] Ronkin considers the function N f defined in by the integral 1 Z log | f (z ,..., z )| dz ··· dz = 1 n 1 n N f (x) n . (1) (2πi) Log−1(x) z1 ··· zn As we will see, the function N f retains some of its properties from the one- variable case, while others are lost or attain a new form. For example, N f is a convex function, but it is no longer piecewise linear. In Section 3 we consider the conse- quences of approximating N f by a piecewise linear function. In Section 5 we investi- −1 gate the relation between local properties of N f and the hypersurface f (0). The properties of the function N f are closely related to the amoeba of f . The amoeba of f , which we denote A f , is defined to be the image in of the hypersurface f −1(0) under the map Log. The term amoeba was first used by Gelfand, Kapranov, and Zelevinsky [5] in the case where f is a polynomial. AMOEBAS, MONGE-AMPEREMEASURES,ANDTRIANGULATIONS` 483 Suppose that E is a connected component of the amoeba complement \ A f . It is not difficult to show that all such components are convex. For example, Log−1(E) is the intersection of Log−1() with the domain of convergence of a certain Laurent series expansion of 1/f , and domains of convergence of Laurent series are always logarithmically convex. In [4] Forsberg, Passare, and Tsikh defined the order of such a component to be the vector ν = (ν1, . , νn) given by the formula 1 Z ∂ f z dz ··· dz = j 1 n ∈ ν j n , x E. (2) (2πi) Log−1(x) ∂z j f (z)z1 ··· zn Here x may be any point in E. They proved, for the case when f is a Laurent poly- nomial, that the order is an integer vector, that is, ν ∈ Zn, that ν is in the Newton polytope of f , and that two distinct components always have different orders. These conclusions remain true, with essentially the same proofs, in the more general setting considered here. Ronkin proved in [12] a theorem that in the language of amoebas amounts to the following statement. THEOREM Let f be a holomorphic function as above. Then N f is a convex function. If U ⊂ is a connected open set, then the restriction of N f to U is affine linear if and only if U does not intersect the amoeba of f . If x is in the complement of the amoeba, then grad N f (x) is equal to the order of the complement component containing x. Sketch of proof The convexity of N f follows from a general theorem because log | f | is plurisub- harmonic (see, e.g., [11, Cor. 1, p. 84]). Differentiation with respect to x j under the integral sign in definition (1) of N f yields precisely the real part of the integral (2) defining the order. However, the integral (2) is always real valued, and this shows im- mediately that N f is affine linear in each connected component of \ A f . The fact that N f is not linear on any open set intersecting the amoeba of f can be proved in several ways. It follows, for instance, from our results in Section 5. 3. Triangulations and polyhedral subdivisions In his doctoral thesis [3], Mikael Forsberg noted that the amoebas of many polynomi- als in two variables have the appearance of slightly thickened graphs. Moreover, there seemed to be some kind of duality between these graphs and certain subdivisions (of- ten triangulations) of the Newton polygon of the polynomial. We make this precise by associating to an amoeba A f a polyhedral complex, which we call its spine. This is done as follows. 484 PASSARE and RULLGARD˚ Assume that f is a holomorphic function defined in (C \{0})n, and write n n A = α ∈ Z ; R \ A f has a component of order α . It may happen that A is empty, but we assume that this is not the case. The most interesting situation arises when A has plenty of points, for example, when the convex hull of A coincides with the Newton polyhedron of f . This always happens if f is a Laurent polynomial. For α ∈ A, let cα = N f (x) − hα, xi, (3) where x is any point in the complement component of order α, and let S(x) = max cα + hα, xi . (4) α∈A This S(x) is a piecewise linear function approximating the Ronkin function. The spine S f is defined as the corner set of S, that is, the set of x where S(x) is nonsmooth. Figure 1. Amoeba of the polynomial + 5 + 2 + 3 2 + 3 4 1 z1 80z1z2 40z1z2 z1z2 (shaded) together with its spine (solid) and the dual triangulation of the Newton polytope We now define a precise meaning to the statement that the spine is dual to a certain subdivision of the convex hull of A. For an example, see Figure 1. Definition 1 Let K be a convex set in Rn. A collection T of nonempty closed convex subsets of K is called a convex subdivision if it satisfies the following conditions. (i) The union of all sets in T is equal to K .
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